Theorem naryrcl 45593

Description: Reverse closure for n-ary (endo)functions. (Contributed by AV, 14-May-2024.)

Hypothesis

Ref	Expression
naryfval.i	⊢ 𝐼 = (0..^𝑁)

Assertion

Ref	Expression
naryrcl	⊢ (𝐹 ∈ (𝑁-aryF 𝑋) → (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V))

Proof of Theorem naryrcl

Dummy variables 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-naryf 45589	. 2 ⊢ -aryF = (𝑥 ∈ ℕ0, 𝑛 ∈ V ↦ (𝑛 ↑m (𝑛 ↑m (0..^𝑥))))
2	1	elmpocl 7425	1 ⊢ (𝐹 ∈ (𝑁-aryF 𝑋) → (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543   ∈ wcel 2112  Vcvv 3398  (class class class)co 7191   ↑_m cmap 8486  0cc0 10694  ℕ₀cn0 12055  ..^cfzo 13203  -aryF cnaryf 45588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-xp 5542  df-dm 5546  df-iota 6316  df-fv 6366  df-ov 7194  df-oprab 7195  df-mpo 7196  df-naryf 45589

This theorem is referenced by:  naryfvalelfv  45594  0aryfvalelfv  45597  fv1arycl  45599  1arymaptfv  45602  fv2arycl  45610  2arymaptfv  45613